1. Superposition of Waves

2. Superposition of Waves
Identical waves in opposite directions:
“standing waves”
2 waves at slightly different frequencies:
“beats”
2 identical waves, but not in phase:
“interference”

3. Principle of Superposition
2 Waves In The Same Medium:
The observed displacement y(x,t) is the algebraic sum of the individual displacements:
y(x,t) =y1(x,t) + y2(x,t)
(for a “linear medium”)

4. What’s Special about Harmonic Waves?
2 waves, of the same amplitude, same angular frequency and wave number (and therefore same wavelength) traveling in the same direction in a medium but are out of phase:
Trig: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2]
Result:
Resultant amplitude

5. Assume  is positive:
For what phase difference  is the total amplitude 2A?
For what phase difference  is the total amplitude A?
For what phase difference  is the total amplitude 0?
For what phase difference  is the total amplitude A/2?

6. wave 1
wave 2
Resultant: Sine wave, same f, different A, intermediate f

7. Standing Waves Two identical waves traveling toward each other in
the same medium with the same wave velocity
y1 = Aosin(kx – ωt)
y2 = Aosin(kx + ωt)
Total displacement, y(x,t) = y1 + y2

8. Trigonometry :
Then:
This is not a traveling wave! Instead this looks like simple
harmonic motion (SHM) with amplitude depending on
position x along the string.

9. For fixed x,the particle motions are simple harmonic oscillations:
t = 0, T, 2T…. y x
-2A0
t = T/2, 3T/2, …
node
node
node
(no vibration)

10. kL=n π , n=1,2,3… determines allowed wavelengths
Nodes are positions where the amplitude is zero. In
particular there must be a node at each end of the string when the string is fixed at each end.
kL=n π , n=1,2,3… determines allowed wavelengths
Since it follows that 2L=n n , for n=1,2,3….

11. The n-th harmonic has n antinodes
Nodes are ½ wavelength apart.
Antinodes (maximum amplitude) are halfway
between nodes.

12. Since:
and
, is the fundamental frequency
The allowed frequencies on a string
fixed at each end are

13. Practical Setup: Fix the ends, use reflections.
We can think of traveling waves reflecting back and forth from the boundaries, and creating a standing wave. The resulting standing wave must have a node at each fixed end. Only certain wavelengths can meet this condition, so only certain particular frequencies of standing wave will be possible.
example:
(“fundamental mode” n=1)
or first harmonic
node
node L

14. λ2
Second Harmonic λ3
Third Harmonic

15. In this case (a one-dimensional wave, on a string with both ends fixed) the possible standing-wave frequencies are multiples of the fundamental: f1, 2f1, 3f2, etc.
This pattern of frequencies depends on the given boundary conditions. The frequencies satisfy a different rule when one end of the string is free.

16. Problem wave at t=0 and T y x 1.2 m f = 150 Hz
8mm x
1.2 m
f = 150 Hz
Write out y(x,t) for the standing wave.

17. Question m When the mass m is doubled, what happens to
a) the wavelength, and
b) the frequency
of the fundamental standing-wave mode?

18. Example 120 cm Oscillator drives Standing waves with
Constant frequency. m
If the frequency of the standing wave remains constant what happens when the weight (tension) is quadrupled?